As a risk manager, you need to evaluate the downside risk of each portfolio from Question 2 over the next year (2021). In particular, you need to compute the Value at Risk for Portfolio 1, 2, and 3 over a one year period. To do so, you need to do the following steps:

1. Using the back-testing results, you need to calibrate the price path for each portfolio, i.e. compute p and p for p = 1, 2, 3 using the daily returns in the OUT period. Report the results in a 2 3 table.

2. Starting with F0,p = 100, the value of Portfolio p for all p = 1, 2, 3 at time t obeys to a Geometric Brownian Motion (GMB), i.e. Ft,p = F0,p exp (p 2 p 2 )t + pZt). where Zt is a standard Brownian Motion (BM). For each portfolio, you need to simulate N = 1000 paths. Given the simulated paths, provide a distribution plot, e.g. boxplot or histogram, for each portfolio. Provide a couple of insights

3. Whats the expected value of each portfolio one year from now?

4. With 95% level of confidence, what is the Value-at-Risk, i.e. V aR(0.05), for each portfolio?

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